Lemma 33.20.4. Let $k$ be a field. Let $f : X \to Y$ be a morphism of locally algebraic $k$-schemes.

For $y \in Y$, the fibre $X_ y$ is a locally algebraic scheme over $\kappa (y)$ hence all the results of Lemma 33.20.3 apply.

Assume $X$ is irreducible. Set $Z = \overline{f(X)}$ and $d = \dim (X) - \dim (Z)$. Then

$\dim _ x(X_{f(x)}) \geq d$ for all $x \in X$,

the set of $x \in X$ with $\dim _ x(X_{f(x)}) = d$ is dense open,

if $\dim (\mathcal{O}_{Z, f(x)}) \geq 1$, then $\dim _ x(X_{f(x)}) \leq d + \dim (\mathcal{O}_{Z, f(x)}) - 1$,

if $\dim (\mathcal{O}_{Z, f(x)}) = 1$, then $\dim _ x(X_{f(x)}) = d$,

For $x \in X$ with $y = f(x)$ we have $\dim _ x(X_ y) \geq \dim _ x(X) - \dim _ y(Y)$.

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